These
descriptions of the real numbers are not sufficiently rigorous by the
modern standards of pure mathematics. The discovery of a suitably
rigorous definition of the real numbers — indeed, the realization that a
better definition was needed — was one of the most important
developments of 19th century mathematics. The currently standard
axiomatic definition is that real numbers form the unique completetotally orderedfield(R,+,·,<),up toisomorphism,[1] Whereas popular constructive definitions of real numbers include declaring them as equivalence classes of Cauchy sequences of rational numbers, Dedekind cuts,
or certain infinite "decimal representations", together with precise
interpretations for the arithmetic operations and the order relation.
These definitions are equivalent in the realm of classical mathematics.